text: For a given nonlinear optical process, the polarization  which generates the output is given by
  where  is the th order nonlinear susceptibility, for .
  It is worth noting that all the even order susceptibilities become zero in centrosymmetric media. A proof of this is as follows.
  Let  be the inversion operator, defined by  for some arbitrary vector . Then applying  to the left and right hand side of the polarization equation above gives
  Adding together this equation with the original polarization equation then gives
  which implies  for  in centrosymmetric media. Q.E.D.
  [Note 1: The final equality can be proven by mathematical induction, by considering two cases in the inductive step; where  is odd and  is even.]
  [Note 2: This proof holds for the case where  is even. Setting  gives the odd case and the remainder of the proof is the same.]
  As a second-order nonlinear process, SFG is dependent on the 2nd order susceptibility , which is a third rank tensor. This limits what samples are accessible for SFG.  Centrosymmetric media include the bulk of gases, liquids, and most solids under the assumption of the electric-dipole approximation, which neglects the signal generated by multipoles and magnetic moments.  At an interface between two different materials or two centrosymmetric media, the inversion symmetry is broken and an SFG signal can be generated.  This suggests that the resulting spectra represent a thin layer of molecules.  A signal is found when there is a net polar orientation.